Biharmonic Riemannian Submersion

• Biharmonic Riemannian submersions are smooth maps that are critical points of the bienergy functional, extending the concept of harmonic maps.
• Key rigidity theorems demonstrate that in constant curvature settings, a biharmonic submersion must be harmonic, emphasizing stringent variational constraints.
• Analytical techniques using adapted frames and integrability data reduce the governing second-order PDEs, clarifying the submersion’s geometric structure.

A biharmonic Riemannian submersion is a smooth Riemannian submersion map between Riemannian manifolds that is a critical point of the bienergy functional, generalizing the concept of harmonic morphisms and connecting the variational theory of harmonic and biharmonic maps to submersion geometry. Formally, if $\varphi:(M,g)\to(N,h)$ is a Riemannian submersion, it is biharmonic if its bitension field vanishes: $$\tau_2(\varphi) := \operatorname{Trace}_g \left( \nabla^{\varphi} \nabla^{\varphi} \tau(\varphi) - R^N(d\varphi, \tau(\varphi))d\varphi \right) = 0.$$ This theory is regarded as the dual counterpart to the well-studied field of biharmonic submanifolds (biharmonic isometric immersions), and plays a central role in the geometric analysis of higher-order elliptic PDEs, the study of rigidity, and the classification of geometric structures under variational constraints.

1. Historical Context and Theoretical Foundations

The foundational development of biharmonic Riemannian submersions is inseparable from the broader history of biharmonic map theory, initiated by the works of B.Y. Chen and G.Y. Jiang on biharmonic submanifolds. The classical result by Chen–Ishikawa and Jiang showed that for isometric immersions $\phi \colon M^2 \to \mathbb{R}^3$ or into real or hyperbolic space forms, the only biharmonic immersions are minimal ones; that is, biharmonicity coincides with harmonicity. This rigidity—absence of proper (i.e., non-harmonic) biharmonic submanifolds—was later harnessed in the submersion setting.

Wang and Ou (Wang et al., 2010) proved the dual statement that any biharmonic Riemannian submersion from a 3-manifold with non-positive constant sectional curvature into a surface is necessarily harmonic. This duality is further formalized and extended: in arbitrary dimensions, a Riemannian submersion $\varphi:(M^{n+1}(c), g)\rightarrow(N^{n}, h)$ from a space of constant sectional curvature $c$ is biharmonic if and only if it is harmonic (Maeta et al., 7 Sep 2025).

These results tie into the general classification program for biharmonic maps. Rigidity properties—including the collapse of biharmonicity to harmonicity for submersions from space forms—contrast with the comparatively richer structure of biharmonic immersions in positively curved ambient spaces. Exceptions to rigidity manifest only in select geometries such as certain BCV spaces or warped products, as explicated in more recent research (Wang et al., 2023).

2. Structure of the Biharmonic Equation for Riemannian Submersions

Given a Riemannian submersion $\varphi:(M,g)\to(N,h)$ from an $(n+1)$-manifold to an $n$-manifold, the tension field is determined by the mean curvature vector field $\mu$ of the fibers: $\tau(\varphi) = -(m-n)\, d\varphi(\mu).$ The biharmonic equation is then

$$\tau_2(\varphi) = - (m-n) d\varphi \left( \sum_{i=1}^n \nabla_{e_i}^M \nabla_{e_i}^M \mu \right) - (m-n)\sum_{i=1}^n \operatorname{Ric}^N (d\varphi(\mu), d\varphi(e_i)) + \text{additional terms},$$

where $\{e_i\}$ is a local orthonormal frame of the horizontal distribution. In the case of 1-dimensional fibers—relevant to the classification from $M^{n+1}$ to $N^n$—the Lie bracket structure of an adapted orthonormal frame leads to scalar equations for the so-called "integrability data." These functions, $f_{ij}^k$, $\kappa_i$, and $\sigma_{ij}$, are determined by

$$[e_i, e_{n+1}] = \kappa_i \, e_{n+1}, \qquad [e_i, e_j] = f_{ij}^k e_k - 2\sigma_{ij} e_{n+1}$$

for $i,j=1,\ldots,n$ and the vertical vector $e_{n+1}$. The biharmonic condition becomes a system of second-order PDEs in these integrability functions and their derivatives, with additional terms involving the curvature tensor of the base.

Householder transformations and orthonormal changes of basis are employed to select a frame in which as many components of $(\kappa_i)$ and $(\sigma_{ij})$ vanish as possible, simplifying the tension and biharmonic equations. Ultimately, for space forms, this reduction shows that the only possibility is vanishing tension field.

3. Rigidity and Classification Results

The main rigidity theorem holds:

Let $\varphi: (M^{n+1}(c), g) \rightarrow (N^{n}, h)$ be a Riemannian submersion from a space form of constant sectional curvature $c$. Then $\varphi$ is biharmonic if and only if it is harmonic (Maeta et al., 7 Sep 2025).

For $n+1=3$, (Wang et al., 2010) provides a full proof: after choosing an adapted frame so that, say, $k_2=0$, the system reduces to

$-\Delta^M k_1 + k_1(-K_N + f_2^2) = 0,$

with $k_1, k_2$ expressing the vertical curvatures from the bracket relations. In dimension $n+1$, the method generalizes: the (many) integrability data must be constant along fibers, allowing further reduction. If any $k_i\neq 0$, the system leads to contradiction, thus $k_i=0$ for all $i$, making the map harmonic.

Beyond space forms, classification work (Wang et al., 2023, Wang et al., 2023) reveals that proper biharmonic Riemannian submersions may exist only in highly restricted contexts, notably:

• From $H^2\times \mathbb{R}$ (or certain BCV spaces) to $\mathbb{R}^2$,
• From the universal covering of $SL(2,\mathbb{R})$ to $\mathbb{R}^2$, where construction of properly biharmonic examples is possible. For Berger spheres (Wang et al., 2023) and Sol geometry (Wang et al., 2023), even these exceptions disappear: every biharmonic submersion is harmonic, or none exist at all.

The table summarizes key existence results:

Domain Geometry	Target	Proper Biharmonic Submersions Exist?
Space forms (all $c$)	Any surface	No
BCV: $H^2(4m)\times\mathbb{R}$	$\mathbb{R}^2$	Yes
BCV: $\widetilde{SL}(2,\mathbb{R})$	$\mathbb{R}^2$	Yes
Berger 3-sphere	Any surface	No
Sol space	Any surface	No
Product $M^2\times\mathbb{R}$	Surface	Only for $M^2$ negatively curved

4. Analytic Aspects and Techniques

The structural reduction of the biharmonic equation for submersions applies tools including:

• Detailed analysis of the adapted orthonormal frame and calculation of the Lie bracket functions,
• Use of symmetry and integrability conditions to ensure that all non-harmonic solutions are ruled out,
• Employment of divergence and Laplacian computations, often yielding scalar elliptic PDEs for crucial integrability data,
• Householder transformations and basis changes to diagonalize the geometric conditions wherever possible (Maeta et al., 7 Sep 2025).

An important technical conclusion: in constant curvature, all integrability data functions are constant along the fiber direction. This allows the biharmonicity condition to be reduced pointwise to an algebraic equation in the tension field components, which then must vanish everywhere, enforcing harmonicity.

5. Relation to Harmonic Maps, Submersions, and Biharmonic Submanifolds

Biharmonic Riemannian submersions sit within the general varitional framework of harmonic and biharmonic maps. Riemannian submersions are harmonic if and only if their fibers are minimal. The study of their biharmonicity investigates a higher-order variational problem—critical points of bienergy. However, in contrast to the broad existence of proper biharmonic immersions in positively curved ambient spaces, the submersion case often exhibits vanishing phenomena for the bitension field only when the tension field is already zero.

This duality is part of a systematic program tracing rigidity phenomena: under nonpositive curvature or high symmetry, proper biharmonic objects are typically excluded, both for immersions/minimal submanifolds and for submersions/fiber-minimal maps (Wang et al., 2010, Maeta et al., 7 Sep 2025, Wang et al., 2023).

6. Examples, Exceptions, and Further Developments

Constructive exceptions—where proper biharmonic Riemannian submersions do occur—have been classified. For BCV spaces and certain warped product settings (Wang et al., 2023, Wang et al., 2023), explicit expressions for the metric and projection reveal nontrivial integrability data satisfying the biharmonicity system, yet with nonzero tension field. In these cases, the fibers are neither harmonic nor totally geodesic, and the higher-order balance encoded in the bitension field is achieved through geometric interplay between warping, curvature, and submersion structure.

Recent developments include:

• The study of $f$-biharmonic submersions, generalizing the energy functional by introducing a weight function $f$, leading to new nontrivial solutions even in otherwise rigid contexts (Wang et al., 2023),
• Complete classification of proper biharmonic submersions from product and warped product manifolds, emphasizing the interplay between fiber geometry, base curvature, and warping functions,
• The non-existence of proper biharmonic submersions from special geometries such as the Berger sphere and Sol geometry, highlighting the sensitivity of the theory to the underlying curvature and symmetry (Wang et al., 2023, Wang et al., 2023).

7. Implications and Research Directions

The theory of biharmonic Riemannian submersions provides sharp illustrations of rigidity and structure in geometric analysis. The explicit reduction and classification in terms of integrability data, together with the absence of proper biharmonic submersions in space forms and homogeneous spaces of high symmetry, underscore the decisive role played by ambient and fiber geometry. Open research directions include:

• Analysis in higher codimension or under relaxed smoothness/homogeneity conditions,
• Examination of $f$-biharmonic submersions and other generalizations involving weighted energies,
• Exploration of moduli and stability properties,
• Extension to non-Riemannian, pseudo-Riemannian, or sub-Riemannian contexts.

This body of results consolidates biharmonic Riemannian submersions as critical elements in the broader theory of higher-order variational geometry, with continuing influence from dualities, classification programs, and explicit geometric constructions (Wang et al., 2010, Maeta et al., 7 Sep 2025, Wang et al., 2023, Wang et al., 2023, Ou, 21 Jul 2024).